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Theorem acneq 8425
Description: Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
acneq  |-  ( A  =  C  -> AC  A  = AC  C )

Proof of Theorem acneq
Dummy variables  f 
g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2539 . . . 4  |-  ( A  =  C  ->  ( A  e.  _V  <->  C  e.  _V ) )
2 oveq2 6293 . . . . 5  |-  ( A  =  C  ->  (
( ~P x  \  { (/) } )  ^m  A )  =  ( ( ~P x  \  { (/) } )  ^m  C ) )
3 raleq 3058 . . . . . 6  |-  ( A  =  C  ->  ( A. y  e.  A  ( g `  y
)  e.  ( f `
 y )  <->  A. y  e.  C  ( g `  y )  e.  ( f `  y ) ) )
43exbidv 1690 . . . . 5  |-  ( A  =  C  ->  ( E. g A. y  e.  A  ( g `  y )  e.  ( f `  y )  <->  E. g A. y  e.  C  ( g `  y )  e.  ( f `  y ) ) )
52, 4raleqbidv 3072 . . . 4  |-  ( A  =  C  ->  ( A. f  e.  (
( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
)  e.  ( f `
 y )  <->  A. f  e.  ( ( ~P x  \  { (/) } )  ^m  C ) E. g A. y  e.  C  ( g `  y
)  e.  ( f `
 y ) ) )
61, 5anbi12d 710 . . 3  |-  ( A  =  C  ->  (
( A  e.  _V  /\ 
A. f  e.  ( ( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
)  e.  ( f `
 y ) )  <-> 
( C  e.  _V  /\ 
A. f  e.  ( ( ~P x  \  { (/) } )  ^m  C ) E. g A. y  e.  C  ( g `  y
)  e.  ( f `
 y ) ) ) )
76abbidv 2603 . 2  |-  ( A  =  C  ->  { x  |  ( A  e. 
_V  /\  A. f  e.  ( ( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
)  e.  ( f `
 y ) ) }  =  { x  |  ( C  e. 
_V  /\  A. f  e.  ( ( ~P x  \  { (/) } )  ^m  C ) E. g A. y  e.  C  ( g `  y
)  e.  ( f `
 y ) ) } )
8 df-acn 8324 . 2  |- AC  A  =  { x  |  ( A  e.  _V  /\  A. f  e.  ( ( ~P x  \  { (/)
} )  ^m  A
) E. g A. y  e.  A  (
g `  y )  e.  ( f `  y
) ) }
9 df-acn 8324 . 2  |- AC  C  =  { x  |  ( C  e.  _V  /\  A. f  e.  ( ( ~P x  \  { (/)
} )  ^m  C
) E. g A. y  e.  C  (
g `  y )  e.  ( f `  y
) ) }
107, 8, 93eqtr4g 2533 1  |-  ( A  =  C  -> AC  A  = AC  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   A.wral 2814   _Vcvv 3113    \ cdif 3473   (/)c0 3785   ~Pcpw 4010   {csn 4027   ` cfv 5588  (class class class)co 6285    ^m cmap 7421  AC wacn 8320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6288  df-acn 8324
This theorem is referenced by:  acndom  8433  dfacacn  8522  dfac13  8523
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